Volume 61 Issue 2 (2011) - Годишник на ТУ - София - Технически ...

FirstIterationSecondIterationx 0 0 x 1 0. 4 x 2 0. 8 x 3 1. 2 x 4 / 2a 0 0.5547 a 1 0. 5449 a 2 1. 1373 a 3 0. 5993 h 0. 0049max | | 0.0067x[0, / 2]x 0 0 x 1 0. 2653 x2 0. 8123 x 3 1. 2205 x 4 / 2a 0 0.5538 a 1 0. 5427 a2 1. 1397 a 3 0. 5858 h 0. 0056max | | 0.0056x[0, / 2]Two iteration proved sufficient, because for accuracy to the fourth decimal place wehave h max x[ 0, / 2]| | . Similarly, if you start Chaikin scheme for initial data obtainedfrom last (second) iteration (i.e. fi ai| i 0,1,2,3 ) and all other f 0i are0zeros, the "memorized" function is obtained. Numerical experiments show that in therange [ 0, / 2]error in "memorizing" of sin(x ) with polynomials of third degree (i.e.polynomial with four coefficients) is 0.0013. The error of "memorizing" of sin(x )with Chaikin subdivision scheme and four starting points is 0.0056. Obviously it isbetter, but the advantage of the proposed method is in its extreme simplicity. It usesonly two mathematical operations addition and multiplication, moreovermultiplicationsare only by 1/4 and 3/4.1.00.80.60.40.20.5 1.0 1.5Fig.1xExample 2. "Memorizing" function f ( x) e in the interval [ 1,1 ].The generalized polynomial used to approximate f (x)in this case isp( x) a00 ( x) a10 ( x 1) a20 ( x 2) a30 ( x 3)The remaining procedure is the same as in Example 1. The numerical results areFirst x 0 1 x 1 0. 5 x 2 0 x 3 0. 5 x 4 1Iteration a 0 0.1863 a 1 0. 5413 a 2 1. 4505 a 3 3. 9779 h 0. 00407max | | 0.016SecondIterationx[1,1]x 0 1 x1 0. 2889 x2 0. 1724 x 3 0. 7614 x 4 1a 0 0.1649 a 1 0. 5489 a 2 1. 4475 a 3 3. 9672 h 0. 011max | | 0.011x[1,1]If we start Chaikin scheme for initial data obtained from last (second) iteration (i.e.0fi ai| i 0,1,2,3 and all other f 0i are zeros, the "memorized" function is ob-43